Convergence to equilibrium for the discrete coagulation-fragmentation equations with detailed balance

TL;DR

This paper proves that solutions to the discrete coagulation-fragmentation equations with detailed balance converge to a known equilibrium, revealing a critical mass that influences long-term behavior, and provides convergence rates.

Contribution

It extends previous results by identifying equilibrium and critical mass for general coefficients under detailed balance, with a new entropy-based convergence proof.

Findings

Solutions converge to equilibrium under detailed balance.

Existence of a critical mass affecting solution behavior.

Quantitative convergence rates are established.

Abstract

Under the condition of detailed balance and some additional restrictions on the size of the coefficients, we identify the equilibrium distribution to which solutions of the discrete coagulation-fragmentation system of equations converge for large times, thus showing that there is a critical mass which marks a change in the behavior of the solutions. This was previously known only for particular cases as the generalized Becker-D\"oring equations. Our proof is based on an inequality between the entropy and the entropy production which also gives some information on the rate of convergence to equilibrium for solutions under the critical mass.